Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
had need for a stub for local diffeomorphism
Suppose a smooth function $p \colon X \to \mathbb{R}^n$ from a diffeological space $X$ to Cartesian space induces at each point an isomorphism on tangent vectors as well as on all higher jets.
Then what sensible extra conditions does it take to conclude that $p$ is in fact a local diffeomorphism, i.e. restricts to a diffeomorphism around an open neighbourhood of each point?
Here I mean tangents and jets defined by equivalence classes of smooth maps into $X$.
Unless I’m missing something, but your condition implies that the Jacobian map $T p \colon TX \to T \mathbb{R}^n$ is full rank and even invertible. The inverse function theorem then guarantees that $p$ is a local diffeomorphism about any point $x\in X$ that has an $n$-manifold neighborhood. Is your question then about $X$’s that at some points fail to be $n$-manifold? But then, it seems to me, that essentially by definition there cannot be a local diffeomorphism from any neighborhood of such a point into $\mathbb{R}^n$.
If you only know that $X$ is a diffeological space and that it has a map $X \to \mathbb{R}^n$ which is an iso on all tangents, what else does it need (if anything) to conclude that $X$ is a manifold sitting by a local diffeomorphism over $\mathbb{R}^n$?
What happened with moneomorphism and epiomorphism? I know that the terms are used rarely, especially outside of Eastern Europe, and especially the second term, but I remember writing about that terminology in nlab and this seemingly completely vanished. nLab search and google show no hits at nLab about those. Did I dream about writing it ?
1 to 5 of 5